Can you explain these questions and how you got the answers?

B

A

Step-by-step explanation:

To find f'(x), take the derivative within each interval.  Clearly, it's 5 in the first one, 2x in the second one, and 4 in the third one.  But we need to determine if the derivative exists at the ends of each interval.

For a derivative to exist at a point x=a, the function must be continuous (f(a⁻) = f(a⁺)), and smooth (f'(a⁻) = f'(a⁺)).

Let's look at x=4.  On the left side:

f(4⁻) = 14, f'(4⁻) = 5

On the right side:

f(4⁺) = 14, f'(4⁺) = 8

So the function is continuous, but not smooth.  Therefore, the derivative doesn't exist there.

Let's try again with x=6.  On the left side:

f(6) = 34, f'(6) = 12

On the right side:

f(6⁺) = 34, f'(6⁺) = 4

Again, the function is continuous, but not smooth, so the derivative doesn't exist there either.

Therefore, f'(x) is:

When we graph f(x) = (4x² − 1) / (x² − 9), we see f(x) has a horizontal tangent line at x=0.

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